Used when people or things follow a sequence: taking turns, doing something in a fixed order, or handling items sequentially.
I arranged the books in order of size.
Please call the names in order.
一人ずつ順に部屋に入ってください。
Please enter the room one by one, in turn.
The teacher called out the names from the roll in order.
I answered the questions one by one, in order.
順に is a general term for doing things in order, while 順々に emphasizes a step-by-step, sequential progression through each item.
順次 is a formal, often written adverb for proceeding in order, while 順に is a neutral, everyday way to say the same thing.
順に means proceeding in the normal/set order, while 逆に means doing the same thing in reverse order.
From 順 (order, sequence) + adverbial particle に. The kanji 順 is a phono-semantic compound: 川 (river) + 頁 (head), originally meaning 'to follow the flow'.